Is it possible to tell Qiskit to execute a parameterized gate where the parameter varies between shots?
Let's say, for whatever reason, I'd like to speed up the dephasing of a qubit during a run. I can accomplish this by explicating implementing the dephasing Kraus operators by using an $R_x(\phi)$ gate, a controlled $Z$ gate and an ancilla qubit. However, this adds a decent amount of complexity, and thus error, to my circuit.
A more clever way to do it is to add an $R_z(\theta)$ gate into the circuit, where $\theta$ is drawn from a normal distribution and thus varies shot to shot. This would mean the circuit would need to be reassembled from the parameterized transpilation each shot, which makes me think it's probably not possible. But I wanted to ask just in case.
I can always do a series of single shot experiments, but I believe the backend (not to mention other users) might have an issue with me submitting thousands of single shot jobs.
So is there any way to implement a circuit which has a parameter which varies from shot to shot?